Suppose $K$ is a quadratic imaginary field, and $\phi:G_K\rightarrow \overline{\mathbb{Q}}$ is a finite order Galois character satisfying $\phi\phi^c=1$ (where $c$ is a complex conjugation). Can we necessarily write $\phi=\psi^c/\psi$ for some finite order character $\psi:G_K\rightarrow \overline{\mathbb{Q}}$? What about for a CM field?